30 research outputs found
Edge-disjoint rainbow trees in properly coloured complete graphs
A subgraph of an edge-coloured complete graph is called rainbow if all its edges
have different colours. The study of rainbow decompositions has a long history,
going back to the work of Euler on Latin squares. We discuss three problems
about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth
Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture.
The main result which we discuss is that in every proper edge-colouring of Kn there
are 10−6n edge-disjoint isomorphic spanning rainbow trees. This simultaneously
improves the best known bounds on all these conjectures. Using our method it is also
possible to show that every properly (n−1)-edge-coloured Kn has n/9 edge-disjoint
spanning rainbow trees, giving a further improvement on the Brualdi-Hollingsworth
Conjectur
A generalization of heterochromatic graphs
In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and
sufficient condition for edge-colored graphs to have a heterochromatic spanning
tree, where a heterochromatic spanning tree is a spanning tree whose edges have
distinct colors. In this paper, we propose -chromatic graphs as a
generalization of heterochromatic graphs. An edge-colored graph is
-chromatic if each color appears on at most edges. We also
present a necessary and sufficient condition for edge-colored graphs to have an
-chromatic spanning forest with exactly components. Moreover, using this
criterion, we show that a -chromatic graph of order with
has an -chromatic spanning forest with exactly
() components if for any
color .Comment: 14 pages, 4 figure
Quaternionic 1-Factorizations and Complete Sets of Rainbow Spanning Trees
A 1-factorization F of a complete graph K2n
is said to be G-regular, or regular under G, if G is an automorphism group of F
acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (Eur J Comb 6:45–48, 1985) on cyclic groups and it is still open when n is even, although several classes of groups were tested in the recent past. It has been recently proved, see Rinaldi (Australas J Comb 80(2):178–196, 2021) and Mazzuoccolo et al. (Discret Math 342(4):1006–1016, 2019), that a G-regular 1-factorization, together with a complete set of rainbow spanning trees, exists for each group G of order 2n, n odd. The existence for each even n>2
was proved when either G is cyclic and n is not a power of 2, or when G is a dihedral group. Explicit constructions were given in all these cases. In this paper we extend this result and give explicit constructions when n>2
is even and G is either abelian but not cyclic, dicyclic, or a non cyclic 2-group with a cyclic subgroup of index 2